On metastability

Résumé

Consider finite state space irreducible and absorbing Markov processes. A general spectral criterion is provided for the absorbing time to be close to an exponential random variable, whatever the starting point. When exiting points are added to the state space, our criterion also insures that the exit time and position are almost independent. Since this is valid for any exiting extension of the state space, it corresponds to an instance of the metastability phenomenon. Simple examples at small temperature suggest that this new spectral criterion is quite sharp. But the main interest of the underlying quantitative approach, based on Poisson equations, is that it does not rely on a small parameter such as temperature, nor on reversibility.